Graphitization by Metal Particles

Graphitization of carbon offers a promising route to upcycle waste biomass and plastics into functional carbon nanomaterials for a range of applications including energy storage devices. One challenge to the more widespread utilization of this technology is controlling the carbon nanostructures formed. In this work, we undertake a meta-analysis of graphitization catalyzed by transition metals, examining the available electron microscopy data of carbon nanostructures and finding a correlation between different nanostructures and metal particle size. By considering a thermodynamic description of the graphitization process on transition-metal nanoparticles, we show an energy barrier exists that distinguishes between different growth mechanisms. Particles smaller than ∼25 nm in radius remain trapped within closed carbon structures, while nanoparticles larger than this become mobile and produce nanotubes and ribbons. These predictions agree closely with experimentally observed trends and should provide a framework to better understand and tailor graphitization of waste materials into functional carbon nanostructures.


Derivation of Free Space Around Metallic Sphere
Basic definition of molar volume (̃) where is volume, the number of moles of material, the molar mass of material and density.  from carbon removed around metal particle with total sphere described by radius rtotal. c) Metal-carbon phase increased in size relative to the initial metal particle; assuming the mixed metal-carbon has a greater density than the carbon dissolved resulting in a symmetrical shell of thickness, t, around the mixed S3 particle. This mixed phase is referred to herein as 'carbide' although this is not necessarily a known stable carbide phase.
The 'total volume' shown in Figure S1b can be calculated as a sum from the metal and carbon volumes, where the metal volume is a sphere and the carbon an outer shell.
Thus, the total radius of the metal particle and dissolved carbon can be written as: Considering the two components, combining (Eqn.4) with the definition of metal molar volume gives an expression relating the volumes of the two; the pure metal sphere and the carbon shell.
Rearranging gives an expression for in terms of metal particle radius. This thickness of the empty shell around the carbide particle ( Figure S 1c) is then found by the difference between the radius of the total system and the radius of the expanded carbide phase.
The linear distance from the edge of the particle to the carbon matrix, within which a nanotube can grow is then 2 as illustrated in Figure S (Eqn.14) The molar volumes are calculated from the densities and thermal expansion coefficients listed in Table S 1.
Figure S 2: Schematic of the empty shell, thickness , around a particle facilitating nanotube growth. a) defines the particle radius and thickness of the empty shell around the particle whilst b) illustrates the growth process within the empty volume.

Derivation of Energy Change of Graphitization
To describe the energy change of growing graphitic carbon nanotubes within the carbon matrix we consider four terms: the surface energy of the nanotube created, the strain energy of the nanotube, the energy of graphitization from converting amorphous carbon to graphitic, and the energy cost of removing the carbon hemi-sphere from the metal particle. The outer radius of the growing nanotubes can be defined by the radius of the metal particle, assuming perpendicular growth from the edge of the particle. This assumption would be difficult to justify for CVD grown nanotubes which often show tangential growth of small SWCNTs; however, all TEM images reported of graphitization within a carbon matrix show a close match between the particle size and tube diameter.
The tube's inner diameter can then be defined from conservation of mass. As the particle migrates through the amorphous carbon a volume of amorphous carbon ( ) is converted to graphitic carbon, which has a greater density. Subscripts and are used throughout to denote amorphous and graphitic carbon.
Assuming the outer radius and volume of carbon contained within the nanotube are both determined by the radius of the metal particle; the curvature becomes independent of the nanoparticle size, depending instead on the ratio of carbon density between the amorphous and graphitic carbon.
Another significant factor in the total energy change is the strain energy of rolling graphite into a tube. This can be calculated using the Young's modulus ( ). Treating the tube wall as a solid block of graphite, the moment ( ) required to bend the block through a moment of inertia ( ) is given in terms of modulus ( ) and radius of curvature ( ).
For a block of thickness and length , where will become the length of the carbon nanotube, the moment of inertia ( ) can be written: The work ( ) required to bend such a block through an angle is: Integrating through a full circle and combining the above expressions results it the work required to bend such a block into a cylinder. This key equation was derived by Tibbetts. 10 This is the energy of a single walled tube, the energy per carbon atom can then be found using the area density of carbon ( ) and making the thickness equal to the layer spacing of graphite ( ). 11 The distinction between and is very minor, where refers the to the radius of curvature of a block whereas refers to the specific radius of the carbon nanotube. This form can be used to derive the strain per carbon atom depending on the radius of the nanotube.
Magnin et al. found this constant to be = 2.14 /Å 2 . 12,13 To clarify, the original paper erroneously states as volume density rather than area density; however, a simple resolution of the units and derivation makes this clear. The third consideration for the total free energy change of the metal particle graphitization is the energy of converting amorphous carbon to graphitic carbon. The free energy change per carbon atom for this transformation was previously reported to be ∆ = 0.065 . 14 The volume of amorphous carbon converted is a cylinder described above; from this the number of carbon atoms can be calculated from the molar mass ( ( )) and density ( ) of amorphous carbon.
Finally, we must consider the energy cost of removing the hemi-sphere of carbon bound to the metal particle, required for the nanotube to start growing outwards. The binding energy per carbon atom ( ) was taken from DFT calculations and are shown in Table S2.
Assuming the binding energy is only significant for the first layer, the number of carbon atoms can be calculated from the area of the hemi-sphere and the area density of carbon (̃). Here the radius of the hemisphere is taken to be the outer radius of the metal particle ( ) plus the equilibrium distance between the metal surface and carbon layer ( ); values taken from literature and listed in Table S2. This term calculates the enthalpy of removing the carbon from the metal surface, however neglects the entropic change from this process. An accurate description of the relative populations of all the vibrational micro-states of the carbon hemi-sphere, with and without the metal particles surface potential, is beyond the scope of this work. This omission is a limitation of the current model; however, this entropic contribution is likely to be small.
Combining these four contributions, equations: 15, 25, 26 and 27, produces an approximate free energy change of the system on graphitization in terms of the nanoparticle radius and tube length.

Comparison of Metals and Conditions
Considering different metals it is curious to note nickel produced larger particles (Figure S 7).
Cobalt broadly followed the same trends as nickel, however the sample size was too small to reliably use. Iron particles were generally smaller in the literature reviewed for this work, with an upper quartile of 40 nm compared with almost double from nickel particles. Examining the different carbon morphologies reported relative to the metal used as a catalyst we found nickel produced a greater proportion of nanoribbons and nanotubes from mobile growth ( Figure S 8), 72% compared to just 48% of the samples from iron. However, considering the iron particles were generally smaller in the literature analysed for this report this remains consistent with the thermodynamics of the growth on different metals.
It is impossible from these observations alone to determine whether the particle size or metal is causing this trend; however, studies using different salts have and the thermal stability of the salt to be a key parameter in controlling the size and distribution of particles produced. More thermally stable salts form larger metal particles and crystalline graphite on their surface compared with less thermally stable salts forming nanoparticles. [17][18][19] Therefore, we conclude the particle size is determining the morphology of carbon produced.